Q33E Vibrating Spring with Damping. U... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q33E (page 173)

Vibrating Spring with Damping. Using the model for a vibrating spring with damping discussed in Example $3$
$(a)$ Find the equation of motion for the vibrating spring with damping if $m = 10 kg, b = 60 kg / \sec, k = 250 kg / \sec^{2}, y (0) = 0 . 3 m,$ and $y' (0) = - 0 . 1 m / \sec$ .
$(b)$ After how many seconds will the mass in part $(a)$ first cross the equilibrium point?
$(c)$ Find the frequency of oscillation for the spring system of part $(a)$ .
$(d)$ Compare the results of problems $32$ and $33$ determine what effect the damping has on the frequency of oscillation. What other effects does it have on the solution?

Short Answer

Expert verified

The equation of motion for the vibrating spring with damping is: $y (t) = e^{- 3 t} (0 . 3 \cos (4 t) + 0 . 2 \sin (4 t))$
The mass crosses the equilibrium at $t = 14$ seconds
The frequency of the spring isrole="math" localid="1654850980914" $f = \frac{2}{蟺}$ .
If the damping increases amplitude and frequency decrease.

Step by step solution

Find the value of c1 and c2

From example $3$ ,

$\begin{array}{l} y' (t) = e^{- 3 t} (- 4 c_{1} \sin (4 t) + 4 c_{2} \cos (4 t)) - 3 e^{- 3 t} (c_{1} \cos (4 t) + c_{2} \sin (4 t)) \\ y' (0) = e^{0} (- 4 c_{1} \sin (0) + 4 c_{2} \cos (0)) - 3 e^{0} (c_{1} \cos (0) + c_{2} \sin (0)) \\ 4 c_{2} - 3 (0 . 3) = - 0 . 1 \\ 4 c_{2} = - 0.1 + 0.9 \\ 4 c_{2} = 0.8 \\ c_{2} = 0.2 \end{array}$

Therefore, the solution is $y (t) = e^{- 3 t} (0 . 3 \cos (4 t) + 0 . 2 \sin (4 t))$ .

Find the value of time.

When the spring crosses the equilibrium $y (t) = 0$ , so we have to find the role="math" localid="1654851280316" $t$

$\begin{array}{l} 0 . 3 \cos (4 t) + 0 . 2 \sin (4 t) = 0 \\ 0 . 3 \cos (4 t) = - 0 . 2 \sin (4 t) \\ \tan (4 t) = - \frac{0.3}{0.2} \\ 4 t = - \arctan (1 . 5) \\ t 鈮� \frac{}{} \\ t 鈮� 14 \end{array}$

So, at $t = 14$ seconds the mass crosses the equilibrium

Find the value of frequency.

Here $尾 = 4$ , the frequency of the spring is $f = \frac{4}{2 蟺} = \frac{2}{蟺}$ .

Comparing the problems 32 and 33

$(d) .$ If the damping increases amplitude and frequency decrease.

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Short Answer

Step by step solution

Find the value of c1 and c2

Find the value of time.

Find the value of frequency.

Comparing the problems 32 and 33

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